On 2D Inverse Problems/Zolotarev problem

From Wikibooks, open books for an open world
Jump to: navigation, search

The following rational approximation problem demonstrates how fast the considered discrete models converge to their continuous analogs. The problem of optimal rational approximation of the square root on an interval was solved by Russian mathematician Zolotarev with the closed form solutions in terms of elliptic integrals, see [DIK] & [PP].

The elliptic function

can be defined by the following integral equation:

Let

The optimal rational approximation function of order (k-1,k).

is given by

where

and

Exercise (**). Use the interlacing of zeros property to prove that the function

can be written as a Stieltjes continued fraction.

The optimal approximation converges exponentially fast with the degree of the rational function.